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Troffaes, M. C. M. (2007) ’Decision making under uncertainty using imprecise probabilities,.’, International journal of approximate reasoning., 45 (1). pp. 17-29.
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Durham Research Online
Deposited in DRO: 14 May 2009
Peer-review status of attached file: Peer-reviewed
Publication status of attached file: Accepted for publication
Citation for published item:
Troffaes, M. C. M. (2007) 'Decision making under uncertainty using imprecise probabilities,.', International journal of approximate reasoning, 45 (1). pp. 17-29.
Further information on publisher’s website: http://dx.doi.org/10.1016/j.ijar.2006.06.001
Use policy
The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-profit purposes provided that :
a full bibliographic reference is made to the original source a link is made to the metadata record in DRO
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Matthias C. M. Troffaes. Decision making under uncertainty using imprecise probabilities.International Journal of Approximate Reasoning, 45:17-29, 2007.
Decision Making under Uncertainty using
Imprecise Probabilities
Matthias C. M. Troffaes
Carnegie Mellon University, Department of Philosophy, Baker Hall 135, Pittsburgh, PA 15213, US
Abstract
Various ways for decision making with imprecise probabilities—admissibility, max-imal expected utility, maxmax-imality, E-admissibility, Γ-maximax, Γ-maximin, all of which are well-known from the literature—are discussed and compared. We gener-alize a well-known sufficient condition for existence of optimal decisions. A simple numerical example shows how these criteria can work in practice, and demonstrates their differences. Finally, we suggest an efficient approach to calculate optimal de-cisions under these decision criteria.
Key words: decision, optimality, uncertainty, probability, maximality, E-admissibility, maximin, lower prevision
1 Introduction
Often, we find ourselves in a situation where we have to make some decision d, which we may freely choose from a setDof available decisions. Usually, we do not choose d arbitrarily in D: indeed, we wish to make a decision that performs best according to some criterion, i.e., anoptimal decision. It is commonly assumed that each decisiondinduces a real-valued gainJd: in that case, a decisiondis considered
optimal in D if it induces the highest gain among all decisions in D. This holds for instance if each decision induces a lottery over some set of rewards, and these lotteries form an ordered set satisfying the axioms of von Neumann Morgenstern
[1], or more generally, the axioms of for instance Herstein and Milnor [2], if we wish to account for unbounded gain.
So, we wish to identify the set opt (D) of all decisions that induce the highest gain. Since, at this stage, there is no uncertainty regarding the gains Jd, d ∈ D,
the solution is simply
opt (D) = arg max
d∈D Jd. (1)
Of course, opt (D) may be empty; however, if the set {Jd: d ∈ D} is a compact
subset of R—this holds for instance if D is finite—then opt (D) contains at least one element. Secondly, note that even if opt (D) contains more than one decision, all decisions d in opt (D) induce the same gain Jd; so, if, in the end, the gain is
all that matters, it suffices to identify only one decision d∗ in opt (D)—often, this greatly simplifies the analysis.
However, in many situations, the gains Jd induced by decisions d in D are
influ-enced by variables whose values are uncertain. Assuming that these variables can be modelled through a random variable X that takes values in some set X (the
possibility space), it is customary to consider the gain Jd as a so-calledgamble on
X, that is, we viewJd as a real-valued gain that is a bounded function of X, and
that is expressed in a fixed state-independent utility scale. So, Jd is a bounded
X–R-mapping, interpreted as an uncertain gain: taking decision d, we receive an amount of utilityJd(x) whenxturns out to be the realisation of X. For the sake of
simplicity, we shall assume that the outcomex ofX is independent of the decision
d we take: this is calledact-state independence. What decision should we take? Irrespective of our beliefs about X, a decision d in D is not optimal if its gain gamble Jd is point-wise dominated by a gain gamble Je for some e in D, i.e., if
there is an e in D such that Je(x) ≥ Jd(x) for all x ∈ X and Je(x) > Jd(x) for
at least one x∈ X: choosing e guarantees a higher gain than choosing d, possibly strictly higher, regardless of the realisation of X. So, as a first selection, let us remove all decisions from D whose gain gambles are point-wise dominated (see Berger [3, Section 1.3.2, Definition 5 ff., p. 10]):
opt≥(D) :={d∈D: (∀e∈D)(Je 6≥Jd orJe =Jd)} (2)
where Je ≥ Jd is understood to be point-wise, and Je 6≥ Jd is understood to be
the negation ofJe≥Jd. The decisions in opt≥(D) are calledadmissible, the other
decisions inDare calledinadmissible. Note that we already recover Eq. (1) if there is no uncertainty regarding the gains Jd, i.e., if allJdare constant functions of X.
When do admissible decisions exist? The set opt≥(D) is non-empty if{Jd: d∈D}
Theorem 3 further on). Note that this condition is sufficient, but not necessary. In what follows, we shall try to answer the following question: given additional information about X, how can we further reduce the set opt≥(D) of admissi-ble decisions? The paper is structured as follows. Section 2 discusses the classical approach of maximising expected utility, and explains why it is not always a de-sirable criterion for selecting optimal decisions. Those problems are addressed in Section 3, discussing alternative approaches to deal with uncertainty and optimal-ity, all of which attempt to overcome the issues raised in Section 2, and all of which are known from the literature. Finally, Section 4 compares these alternative approaches, and explains how optimal decisions can be obtained in a computation-ally efficient way. A few technical results are deferred to the appendix, where we, among other things, generalize a well-known technical condition on the existence of optimal decisions.
2 Maximising Expected Utility?
In practice, beliefs about X are often modelled by a (possibly finitely additive) probability measure µ on a field F of subsets of X, and one then arrives at a set of optimal decisions by maximising their expected utility with respect to µ; see for instance Raiffa and Schlaifer [4, Section 1.1.4, p. 6], Levi [5, Section 4.8, p. 96, ll. 23–26], or Berger [3, Section 1.5.2, Paragraph I, p. 17]. Assuming that the field F is sufficiently large such that the gains Jd are measurable with respect to
F—this means that everyJdis a uniform limit ofF-simple gambles—the expected utility of the gain gambles Jd is given by:
Eµ(Jd) := Z
Jddµ,
where we take for instance the Dunford integral on the right hand side; see Dun-ford [6, p. 443, Sect. 3], and DunDun-ford and Schwartz [7, Part I, Chapter III, Defini-tion 2.17, p. 112]—this linear integral extends the usual textbook integral (see for instance Kallenberg [8, Chapter 1]) to case where µis not σ-additive. Recall that we have assumed act-state independence: µis independent of d.
As far as it makes sense to rank decisions according to the expected utility of their gain gambles, we shouldmaximise expected utility:
optEµ(D) := arg max
When do optimal solutions exist? The set optEµ(D) is guaranteed to be non-empty if {Jd: d ∈ D} is a non-empty and compact subset of the set L(X) of
all gambles on X, with respect to the supremum norm. Actually, this technical condition is sufficient for existence with regard to all of the optimality conditions we shall discuss further on. Therefore, without further ado, we shall assume that
{Jd: d ∈ D} is non-empty and compact with respect to the supremum norm. A
slightly weaker condition is assumed in Theorem 5, in the appendix of this paper. Unfortunately, it may happen that our beliefs about X cannot be modelled by a probability measure, simply because we have insufficient information to identify the probability µ(A) of every event A in F. In such a situation, maximising expected utility usually fails to give an adequate representation of optimality.
For example, letX be the unknown outcome of the tossing of a coin; say we only
know that the outcome will be either heads or tails (so X = {H, T}), and that the probability of heads lays between 28% and 70%. Consider the decision set
D={1,2,3,4,5,6}and the gain gambles
J1(H) = 4, J1(T) = 0, J2(H) = 0, J2(T) = 4, J3(H) = 3, J3(T) = 2, J4(H) = 12, J4(T) = 3, J5(H) = 4720, J5(T) = 4720, J6(H) = 4110, J6(T) = −103,
Clearly, opt≥(D) = {1,2,3,4,5,6}, and
optEµ(D) = {2}, if µ(H)< 25, {2,3}, if µ(H) = 25, {3}, if 25 < µ(H)< 23, {1,3}, if µ(H) = 23, {1}, if µ(H)> 23.
Concluding, if we haveno additional information aboutX, but still insist on using a particular (and necessarily arbitrary)µ, which is only required to satisfy 0.28≤
µ(H) ≤0.7, we find that optEµ(D) is not very robust against changes in µ. This shows that maximising expected utility fails to give an adequate representation of optimality in case of ignorance about the precise value of µ.
3 Generalising to Imprecise Probabilities
Of course, if we have sufficient information such thatµ can be identified, nothing is wrong with Eq. (3). We shall therefore try to generalise Eq. (3). In doing so, following Walley [9], we shall assume that our beliefs about X are modelled by a real-valued mapping P defined on a—possibly only very small—set Kof gambles, that represents our assessment of thelower expected utility P(f) for each gamblef
inK;1 note thatKcan be chosen empty if we are completely ignorant. Essentially, this means that instead of a single probability measure on F, we now identify a closed convex set M of finitely additive probability measures µ on F, described by the linear inequalities
(∀f ∈ K)(P(f)≤Eµ(f)). (4)
We choose the domainF of the measuresµsufficiently large such that all gambles of interest, in particular those in Kand the gain gambles Jd, are measurable with
respect to F. Without loss of generality, we can assume F to be the power set of
X, although in practice, it may be more convenient to choose a smaller field. For a given F-measurable gamble g, not necessarily in K, we may also derive a lower expected utilityEP(g) by minimisingEµ(g) subject to the above constraints,
and an upper expected utility EP(g) = −EP(−g) by maximising Eµ(g) over the
above constraints. In case X and K are finite, this simply amounts to solving a linear program.
In the literature, M is called a credal set (see for instance Giron and Rios [10], and Levi [5, Section 4.2, pp. 76–78], for more comments on this model), and P
is called a lower prevision (because they generalise the previsions, which are fair prices, of De Finetti [11, Vol. I, Section 3.1, pp. 69–75]).
The mapping EP obtained, corresponds exactly to the so-called natural exten-sion of P (to the set of F-measurable gambles), where P(f) is interpreted as a supremum buying price for f (see Walley [9, Section 3.4.1, p. 136]). In this inter-pretation, for any s < P(f), we are willing to pay any utility s < P(f) prior to observation ofX, if we are guaranteed to receive f(x) once x turns out to be the outcome ofX. The natural extension then corresponds to the highest price we can obtain for an arbitrary gamble g, taken into account the assessed prices P(f) for 1 The upper expected utility of a gamble f is P(f) if and only if the lower expected utility of −f is −P(f). So, for any gamble f in K, P(−f) = −P(f), and therefore, without loss of generality, we can restrict ourselves to lower expected utility.
f ∈ K. Specifically, EP(g) = sup ( α+ n X i=1 λiP(fi) : α+ n X i=1 λifi ≤g ) , (5)
where α varies overR, n overN, λ1, . . . , λn vary overR+, and f1, . . . , fn overK.
It may happen thatMis empty, in which case EP is undefined (the supremum in Eq. (5) will always be +∞). This occurs exactly when P incurs a sure loss as a lower prevision, that is, if we can find a finite collection of gamblesf1, . . . ,fninK
such thatPn
i=1P(fi)>sup [Pni=1fi], which means that we are willing to pay more
for this collection than we can ever gain from it, which makes no sense of course. Finally, it may happen that EP does not coincide with P on K. This points to a form of incoherence in P: this situation occurs exactly when we can find a finite collection of gambles f0, f1, . . . , fn and non-negative real numbers λ1, . . . , λn,
such that α+ n X i=1 λifi ≤f0, but alsoP(f0)< α+ n X i=1 λiP(fi).
This means that we can construct a price forf0, using the assessed pricesP(fi) for
fi, which is strictly higher thanP(f0). In this sense,EP correctsP, as is apparent
from Eq. (5).
Although the belief model described above is not the most general we may think of, it is sufficiently general to model both expected utility and complete ignorance: these two extremes are obtained by takingMeither equal to a singleton, or equal to the set of all finitely additive probability measures on F (i.e., K = ∅). It also allows us to demonstrate the differences between different ways to make decisions with imprecise probabilities on the example we presented before.
In that example, the given information can be modelled by, say, a lower previsionP
onK={IH,−IH}, defined byP(IH) = 0.28 andP(−IH) =−0.7, whereIH is the
gamble defined by IH(H) = 1 and IH(T) = 0. For thisP, the set M corresponds
exactly to the set of all probability measures µ on F = {∅,{H},{T},{H, T}}, such that 0.28≤µ(H)≤0.7. We also easily find for any gamble f on X that
3.1 Γ-Maximin and Γ-Maximax
As a very simple way to generalise Eq. (3), we could take the lower expected utility EP as a replacement for the expected utility Eµ (see for instance Gilboa
and Schmeidler [12], or Berger [3, Section 4.7.6, pp. 215–223]): optE
P (D) := argd∈maxopt≥(D)EP(Jd); (6)
this criterion is called Γ-maximin, and amounts to worst-case optimisation: we take a decision that maximises the worst expected gain. For example, if we consider the decision as a game against nature, who is assumed to choose a distribution inM
aimed at minimizing our expected gain, then the Γ-maximin solution is the best we can do. Applied on the example of Section 2, we find as a solution optE
P(D) ={5}.
In case K=∅, i.e., in case of complete ignorance about X, it holds that EP(f) = infx∈Xf(x). Hence, in that case, Γ-maximin coincides with maximin (see Berger
[3, Eq. (4.96), p. 216]), ranking decisions by the minimal (or infimum, to be more precise) value of their gain gambles.
Some authors consider best-case optimisation, taking a decision that maximises the best expected gain (see for instance Satia and Lave [13]). In our example, the “Γ-maximax” solution is optE
P (D) ={2}.
3.2 Maximality
Eq. (3) is essentially the result of pair-wise preferences based on expected utility: defining the strict partial order >µ on D asd >µe whenever Eµ(Jd)>Eµ(Je), or
equivalently, whenever Eµ(Jd−Je)>0, we can simply write
optEµ(D) = max>µ
opt≥(D),
where the operator max>µ(·) selects the >µ-maximal, i.e., the >µ-undominated elements from a set with strict partial order >µ.
Using the supremum buying price interpretation, it is easy to derive pair-wise preferences from P: define >P as d >P e whenever EP(Jd −Je) > 0. Indeed,
EP(Jd −Je) > 0 means that we are disposed to pay a strictly positive price in
d over e (see Walley [9, Sections 3.9.1–3.9.3, pp. 160–162]). Since >P is a strict
partial order, we arrive at
opt>P (D) :=max>P
opt≥(D)
={d∈opt≥(D) : (∀e∈opt≥(D))(EP(Je−Jd)≤0)} (7)
as another generalisation of Eq. (3), called maximality. Note that >P can also be
viewed as a robustification of>µoverµinM. Applied on the example of Section 2,
we find opt>
P (D) ={1,2,3,5}as a solution.
Note that Walley [9, Sections 3.9.2, p. 161] has a slightly different definition: instead of working from the set of admissible decisions as in Eq. (7), Walley starts with ranking d > e if EP(Jd −Je) > 0 or (Jd ≥ Je and Jd 6= Je), and then selects
those decisions from D that are maximal with respect to this strict partial order. Using Theorem 3 from the appendix, it is easy to show that Walley’s definition of maximality coincides with the one given in Eq. (7) whenever the set {Jd: d∈D}
is weakly compact. This is something we usually assume to ensure the existence of admissible elements; in particular, weak compactness is assumed in Theorem 5 (see appendix). The benefit of Eq. (7) over Walley’s definition is that Eq. (7) is easier to manage in the proofs in the appendix.
3.3 Interval Dominance
Another robustification of >µ is the strict partial ordering AP defined by d AP
e whenever EP(Jd) > EP(Je); this means that the interval [EP(Jd),EP(Jd)] is
completely on the right hand side of the interval [EP(Je),EP(Je)]. The above
ordering is therefore called interval dominance (see Zaffalon, Wesnes, and Petrini [14, Section 2.3.3, pp. 68–69] for a brief discussion and references).
optAP(D) :=maxAP opt≥(D)
={d ∈opt≥(D) : (∀e∈opt≥(D))(EP(Je)≤EP(Jd))} (8)
The resulting notion is weaker than maximality: applied on the example of Sec-tion 2, optAP(D) = {1,2,3,5,6}, which is strictly larger than opt>P(D).
3.4 E-Admissibility
In the example of Section 2, we have shown that optEµ(D) may not be very robust against changes inµ. Robustifying optEµ(D) against changes ofµinM, we arrive at
optM(D) := [
µ∈M
optEµ(D) ; (9)
this provides another way to generalise Eq. (3). The above criterion selects those admissible decisions in D that maximize expected utility with respect to at least one µ in M; i.e., they select the E-admissible (see Good [15, p. 114, ll. 8–9], or Levi [5, Section 4.8, p. 96, ll. 8–20]) decisions among the admissible ones. We find optM(D) ={1,2,3}for the example.
In caseµis defined on℘(X) andµ({x})>0 for allx∈ X, then every E-admissible decision is also admissible, and hence, in that case, optM(D) gives us exactly the set of E-admissible options.
4 Which Is the Right One?
Evidently, it is hard to pinpoint the right choice. Instead, let us ask ourselves: what properties do we want our notion of optimality to satisfy? Let us summarise a few important guidelines.
Clearly, whatever notion of optimality, it seems reasonable to exclude inadmissible decisions. For ease of exposition, let’s assume that the inadmissible decisions have already been removed from D, i.e.,D = opt≥(D); this implies in particular that
optM(D) gives us the set of E-admissible decisions.
Now note that, in general, the following implications hold:
Γ-maximax Γ-maximin E-admissible maximal interval dominance ? H H H H H H HHj ?? -H H H H H H H H j ?
as is also demonstrated by our example. A proof is given in the appendix, Theo-rem 1.
E-admissibility, maximality, and interval dominance have the nice property that the more determinate our beliefs (i.e., the smaller M), the smaller the set of optimal decisions. In contradistinction, Γ-maximin and Γ-maximax lack this prop-erty, and usually only select a single decision, even in case of complete ignorance. However, if we are only interested in the most pessimistic (or most optimistic) solution, disregarding other reasonable solutions, then Γ-maximin (or Γ-maximax) seems appropriate. Utkin and Augustin [16] have collected a number of nice al-gorithms for finding Γ-maximin and Γ-maximax solutions, and even mixtures of these two. Seidenfeld [17] has compared Γ-maximin to E-admissibility, and argued against Γ-maximin in sequential decision problems.
If we do not settle for Γ-maximin (or Γ-maximax), should we choose E-admissibility, maximality, or interval dominance? As already mentioned, interval dominance is weaker than maximality, so in general we will end up with a larger (and arguably too large) set of optimal options. Assuming the non-admissible decisions have been weeded, a decisiond is not optimal inDwith respect to interval dominance if and only if
EP(Jd)<sup e∈D
EP(Je). (10)
Thus, ifD has n elements, interval dominance requires us to calculate 2n natural extensions, and make 2ncomparisons, whereas for maximality, by Eq. (7), we must calculate n2 −n natural extensions, and perform n2 −n comparisons—roughly speaking, each natural extension is a linear program inm(size of X) variables and
r(size ofK) constraints, or vice versa if we solve the dual program. So, comparing maximality and interval dominance, we face a tradeoff between computational speed and number of optimal options.
However, this also means that interval dominance is a means to speed up the cal-culation of maximal and E-admissible decisions: because every maximal decision is also interval dominant, we can invoke interval dominance as a first computa-tionally efficient step in eliminating non-optimal decisions, if we eventually opt for maximality or E-admissibility. Indeed, eliminating those decisions d that sat-isfy Eq. (10), we will also eliminate those decisions that are neither maximal, nor E-admissible.
Regarding sequential decision problems, we note that dynamic programming tech-niques cannot be used when using interval dominance (see De Cooman and Troffaes [18]), and therefore, since dynamic programming yields an exponential speedup, maximality and E-admissibility are certainly preferred over interval dominance
once dynamics enter the picture.
This leaves E-admissibility and maximality. They are quite similar: they coincide on all decision sets D that contain two decisions. In case we consider larger de-cision sets, they coincide if the set of gain gambles is convex (for instance, if we considerrandomised decisions). As already mentioned, E-admissibility is stronger than maximality, and also has some other advantages over maximality. For in-stance, 15J2+45J3 >P J5, so, choosing decision 2 with probability 20% and decision 3 with probability 80% is preferred to decision 5. Therefore, we should perhaps not consider decision 5 as optimal.
E-admissibility is not vulnerable to such argument, since no E-admissible decision can be dominated by randomized decisions: if for some µ ∈ M it holds that
Eµ(Jd−Je)≥0 for all e∈D, then also
Eµ Jd− n X i=1 λiJei ! = n X i=1 λiEµ(Jd−Jei)≥0
for any convex combination Pn
i=1λiJei of gain gambles, and hence, it also holds thatEP(Pn
i=1λiJei−Jd)≤0 which means that no convex combination
Pn
i=1λiJei can dominate Jd with respect to >P.
A powerful algorithm for calculating E-admissible options has been recently sug-gested by Utkin and Augustin [16, pp. 356–357], and independently by Kikuti, Cozman, and de Campos [19, Sec. 3.4]. If D has n elements, finding all (pure) E-admissible options requires us to solve n linear programs in m variables and
r+n constraints.
As we already noted, through convexification of the decision set, maximality and E-admissibility coincide. Utkin and Augustin’s algorithm can also cope with this case, but now one has to consider in the worst casen! linear programs, and usually several less: the worst case only obtains if all options are E-admissible. For instance, if there are only ` E-admissible pure options, one has to consider only at most
`! +n−` of those linear programs, and again, usually less.
In conclusion, the decision criterion to settle for in a particular application, depends at least on the goals of the decision maker (what properties should optimality sat-isfy?), and possibly also on the size and structure of the problem if computational issues arise.
Acknowledgements
I especially want to thank Teddy Seidenfeld for the many instructive discussions about maximality versus E-admissibility. I also wish to thank two anonymous ref-erees for their helpful comments. This paper has been supported by the Belgian American Educational Foundation. The scientific responsibility rests with its au-thor.
A Proofs
This appendix is dedicated to proving the connections between the various op-timality criteria, and existence results mentioned throughout the paper. In the whole appendix, we assume the following:
Recall, D denotes some set of decisions, and every decision d∈ D induces a gain gambleJd∈ L(X), whereL(X) is the set of all gambles (boundedX–Rmappings). P denotes a lower prevision, defined on a subset K of L(X). With F we denote a field on X such that all gain gambles Jd and gambles in K are measurable with
respect toF, i.e., are a uniform limit ofF-simple gambles.F could be for instance the power set of X.
P is assumed to avoid sure loss, and EP is its natural extension to the set of all
F-measurable gambles.Mis the credal set representingP, as defined in Section 3. We will make deliberate use of the properties of natural extension (for instance, superadditivity:EP(f+g)≥EP(f) +EP(g), and hence alsoEP(f−g)≤EP(f)−
EP(g)). We refer to Walley [9, Sec. 2.6, p. 76, and Sec. 3.1.2, p. 123] for an overview and proof of these properties.
We use the symbol µ for an arbitrary finitely additive probability measure onF, and Eµ denotes the Dunford integral with respect toµ. This integral is defined on
A.1 Connections between Decision Criteria
Theorem 1 The following relations hold.
optE
P(D)⊆optM(D)⊆opt>P(D)⊆optAP (D)
optE
P(D)⊆opt>P(D)
PROOF. LetJ ={Jd: d∈D}.
Suppose thatdis Γ-maximax inD:JdmaximisesEP in max≥(J). SinceEP is the
upper envelope ofM, andMis weak-* compact (see Walley [9, Sec. 3.6]), there is aµ inMsuch that EP(Jd) = Eµ(Jd). But, Eµ(Je)≤EP(Je)≤EP(Jd) = Eµ(Jd),
for every Je∈max≥(J) because d is Γ-maximax. Thus,d belongs to optM(D).
Suppose that d ∈ optM(D): there is a µ in M such that Jd maximises Eµ in
max≥(J). But then, because EP is the lower envelope of M, EP(Je − Jd) ≤
Eµ(Je−Jd) = Eµ(Je)−Eµ(Jd) ≤0 for all Je in max≥(J). Hence, by Eq. (7) on
p. 8, d must be maximal.
Suppose that d is maximal. Then, again by Eq. (7), EP(Je−Jd)≤0 for all Je in
max≥(J). But,EP(Je)−EP(Jd)≤EP(Je−Jd), hence, also EP(Je)≤EP(Jd) for
allJe in max≥(J), which means thatd belongs to optAP(D).
Finally, suppose that d is Γ-maximin: Jd maximises EP in max≥(J). But then
EP(Je−Jd)≤EP(Je)−EP(Jd)≤0 for all Je in max≥(J); d must be maximal.
A.2 Existence
We first prove a technical but very useful lemma about the existence of optimal elements with respect to preorders; it’s an abstraction of a result proved by De Cooman and Troffaes [18]. Let’s start with a few definitions.
A preorder is simply a reflexive and transitive relation.
Let V be any set, and let Q be any preorder on V. An element v of a subset S
Q-maximal elements is denoted by max Q(S) := v ∈ S: (∀w∈ S)(wQv =⇒ v Qw) . (A.1)
For any v inS, we also define the up-set of v relative to S as
↑S
Qv :={w∈ S: wQv}.
Lemma 2 Let V be a Hausdorff topological space. Let Q be any preorder on V such that for any v in V, the set ↑V
Qv is closed. Then, for any non-empty compact
subset S of V, the following statements hold. (i) For every v in S, the set ↑S
Qv is non-empty and compact.
(ii) The set max
Q(S) of Q-maximal elements of S is non-empty.
(iii) For every v in S, there is a Q-maximal element w of S such that wQv.
PROOF. (i). SinceQ is reflexive, it follows that v Q v, so↑S
Qv is non-empty. Is it compact? Clearly, ↑S Qv = S ∩ ↑ V Qv, so ↑ S
Qv is the intersection of a compact set and a closed set, and therefore ↑S
Qv must be compact too.
(ii). LetS0 be any subset of the non-empty compact set S that is linearly ordered
with respect toQ. If we can show thatS0has an upper bound in Swith respect to Q, then we can infer from a version of Zorn’s lemma [20, (AC7), p. 144] (which also holds for preorders) that S has a Q -maximal element. Let then {v1, v2, . . . , vn}
be an arbitrary finite subset of S0. We can assume without loss of generality that
v1 Q v2 Q · · · Q vn, and consequently ↑ S Qv1 ⊆ ↑ S Qv2 ⊆ · · · ⊆ ↑ S Qvn. This implies that the intersectionTn
k=1↑
S
Qvk =↑
S
Qv1 of these up-sets is non-empty: the collection
{↑S
Qv: v ∈ S
0} of compact and hence closed (V is Hausdorff) subsets of S has the
finite intersection property. Consequently, since S is compact, the intersection T
v∈S0↑S
Qv is non-empty as well, and this is the set of upper bounds ofS
0 inS with
respect to Q . So, by Zorn’s lemma, S has a Q -maximal element: max
Q(S) is non-empty.
(iii). Combine (i) and (ii) to show that the non-empty compact set ↑S
Qv has a maximal element wwith respect to Q. It is then a trivial step to prove that w is also Q -maximal in S: we must show that for any u in S, if u Q w, then w Q u. But, if u Qw, then also uQ v since w Qv by construction. Hence, u ∈ ↑S
Qv, and since w is Q-maximal in ↑S
Qv, it follows that wQu.
That is, a net fα in L(X) converges weakly to f in L(X) if limαfα(x) =f(x) for
allx∈ X.
Theorem 3 If J ={Jd:d ∈D} is a non-empty and weakly compact set, then D contains at least one admissible decision, and even more, for every decision e in
D, there is an admissible decision d in D such that Jd≥Je.
PROOF. It is easy to derive from Eq. (2) that
opt≥(D) = {d ∈D: (∀e∈D)(Je≥Jd =⇒ Jd ≥Je)}.
Hence, a decision is admissible in D exactly when its gain gamble is ≥-maximal inJ. We must show that J has ≥-maximal elements.
By Lemma 2, it suffices to prove that, for every f ∈ L(X), the set Gf = {g ∈
L(X) :g ≥f} is closed with respect to the topology of point-wise convergence. Let gα be a net inGf, and suppose that gα converges point-wise to g ∈ L(X): for
every x∈ X, limαgα(x) =g(x). But, since gα(x)≥f(x) for every X, it must also
hold that g(x) = limαgα(x) ≥ f(x). Hence, g ∈ Gf. We have shown that every
converging net inGf converges to a point inGf. Thus,Gf is closed. This establishes
the theorem.
Let’s now introduce a slightly stronger topology on L(X). This topology has no particular name in the literature, so let’s just call it theτ-topology. It is determined by the following convergence.
Definition 4 Say that a net fα in L(X) τ-converges to f in L(X), if (i) limαfα(x) =f(x) for all x∈ X (point-wise convergence), and (ii) limαEP(|fα−f|) = 0 (convergence in EP(| · |)-norm).
This convergence induces a topologyτ onL(X): it turnsL(X) into a locally convex topological vector space, which also happens to be Hausdorff. A topological basis at 0 consists for instance of the convex sets
{f ∈ L(X) : EP(|f|)< and f(x)< δ(x)},
for > 0, and δ(x) > 0 for all x ∈ X. It has more open sets and more closed sets than the weak topology, but it has less compact sets than the weak topology. On the other hand, this topology is weaker than the supremum norm topology,
so it has fewer open and closed sets, and more compact sets, compared to the supremum norm topology. Note that in case X is finite, it reduces to the weak topology, which is in that case also equivalent to the supremum norm topology. Note thatEP, EP, and Eµ for all µ∈ M, are τ-continuous, simply because
EP(|fα−f|)≥ |EP(fα)−EP(f)|,
EP(|fα−f|)≥ |EP(fα)−EP(f)|, and
EP(|fα−f|)≥ |Eµ(fα)−Eµ(f)|
(see Walley [9, p. 77, Sec. 2.6.1(l)]). We will exploit this fact in the proof of the following theorem, generalising a result due to Walley [9, p. 161, Sec. 3.9.2].
Theorem 5 If J = {Jd: d ∈ D} is non-empty and compact with respect to the
τ-topology, then the following statements hold. (i) optEµ(D) is non-empty for all µ∈ M. (ii) optE P(D) is non-empty. (iii) optE P(D) is non-empty. (iv) opt> P (D) is non-empty. (v) optA P (D) is non-empty. (vi) optM(D) is non-empty.
PROOF. (i). Introduce the following order on L(X): say that f Q g whenever
Eµ(f) ≥ Eµ(g). Let’s first show that, for all f ∈ L(X), the set Gf = {g ∈
L(X) :g Qf} isτ-closed.
Letgαbe a net inGf, and suppose that gα τ-converges tog ∈ L(X). Since the
inte-gralEµisτ-continuous, it follows thatEµ(g) = limαEµ(gα)≥Eµ(f). Concluding,
g belongs to Gf. We have established that every converging net inGf converges to
a point in Gf. Thus, Gf isτ-closed.
By Lemma 2, it follows that J has at least one Q -maximal element Je, that is,
Je maximisesEµ inJ. Since anyτ-compact set is also weakly compact, there is a
≥-maximal elementJd inJ such thatJd≥Je, by Theorem 3. But then,Eµ(Jd)≥
Eµ(Je), and hence,Jd also maximises Eµ in J. Because Jd is ≥-maximal in J, it
also maximisesEµ in max≥(J). This establishes thatd belongs to optEµ(D): this
set is non-empty.
EP(g). Continue along the lines of (i), using the fact that EP isτ-continuous. (iii). Again along the lines of (i), with f Qg whenever EP(f)≥EP(g).
(iv)&(v)&(vi). Immediate, by (iii) and Theorem 1.
References
[1] J. von Neumann, O. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, 1944.
[2] I. N. Herstein, J. Milnor, An axiomatic approach to measurable utility, Econometrica 21 (2) (1953) 291–297.
[3] J. O. Berger, Statistical Decision Theory and Bayesian Analysis, 2nd Edition, Springer, 1985.
[4] H. Raiffa, R. Schlaifer, Applied Statistical Decision Theory, MIT Press, 1961. [5] I. Levi, The Enterprise of Knowledge. An Essay on Knowledge, Credal Probability,
and Chance, MIT Press, Cambridge, 1983.
[6] N. Dunford, Integration in general analysis, Transactions of the American Mathematical Society 37 (3) (1935) 441–453.
[7] N. Dunford, J. T. Schwartz, Linear Operators, John Wiley & Sons, New York, 1957. [8] O. Kallenberg, Foundations of Modern Probability, 2nd Edition, Probability and Its
Applications, Springer, 2002.
[9] P. Walley, Statistical Reasoning with Imprecise Probabilities, Chapman and Hall, London, 1991.
[10] F. J. Giron, S. Rios, Quasi-Bayesian behaviour: A more realistic approach to decision making?, in: J. M. Bernardo, J. H. DeGroot, D. V. Lindley, A. F. M. Smith (Eds.), Bayesian Statistics, University Press, Valencia, 1980, pp. 17–38.
[11] B. De Finetti, Theory of Probability: A Critical Introductory Treatment, Wiley, New York, 1974–5, two volumes.
[12] I. Gilboa, D. Schmeidler, Maxmin expected utility with non-unique prior, Journal of Mathematical Economics 18 (2) (1989) 141–153.
[13] J. K. Satia, J. Roy E. Lave, Markovian decision processes with uncertain transition probabilities, Operations Research 21 (3) (1973) 728–740.
[14] M. Zaffalon, K. Wesnes, O. Petrini, Reliable diagnoses of dementia by the naive credal classifier inferred from incomplete cognitive data, Artificial Intelligence in Medicine 29 (1–2) (2003) 61–79.
[15] I. J. Good, Rational decisions, Journal of the Royal Statistical Society, Series B 14 (1) (1952) 107–114.
[16] L. V. Utkin, T. Augustin, Powerful algorithms for decision making under partial prior information and general ambiguity attitudes, in: F. G. Cozman, R. Nau, T. Seidenfeld (Eds.), Proceedings of the Fourth International Symposium on Imprecise Probabilities and Their Applications, 2005, pp. 349–358.
[17] T. Seidenfeld, A contrast between two decision rules for use with (convex) sets of probabilities: Gamma-maximin versus E-admissibility, Synthese 140 (1–2) (2004) 69–88.
[18] G. de Cooman, M. C. M. Troffaes, Dynamic programming for deterministic discrete-time systems with uncertain gain, International Journal of Approximate Reasoning 39 (2–3) (2004) 257–278.
[19] D. Kikuti, F. G. Cozman, C. P. de Campos, Partially ordered preferences in decision trees: Computing strategies with imprecision in probabilities, in: R. Brafman, U. Junker (Eds.), Multidisciplinary IJCAI-05 Workshop on Advances in Preference Handling, 2005, pp. 118–123.
[20] E. Schechter, Handbook of Analysis and Its Foundations, Academic Press, San Diego, 1997.
